For s>5/2, a polynomial derivative Schrödinger nonlinearity is locally well-posed in H^s on the torus exactly when the imaginary part of the derivative of the nonlinearity with respect to ∂_x u has zero mean for every datum.
Chihara, The initial value problem for Schr¨ odinger equations on the torus, Int
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Well- and ill-posedness of the Cauchy problem for semi-linear Schr\"odinger equations on the torus
For s>5/2, a polynomial derivative Schrödinger nonlinearity is locally well-posed in H^s on the torus exactly when the imaginary part of the derivative of the nonlinearity with respect to ∂_x u has zero mean for every datum.